Specific Energy & Critical Depth — Reference
Specific energy is energy per unit weight relative to the channel bottom. It controls open-channel transitions, controls, and the sub/supercritical state of the flow.
Core Equations
Flow State by Depth
| Condition | State | Character |
|---|---|---|
| y > yc (Fr < 1) | Subcritical | Tranquil, deep/slow; controls act from downstream |
| y = yc (Fr = 1) | Critical | Minimum E; unstable, used as a flow-measurement control |
| y < yc (Fr > 1) | Supercritical | Rapid, shallow/fast; controls act from upstream |
Alternate Depths
For any E > Emin, two alternate depths pass the same Q at the same specific energy — one subcritical, one supercritical. They are equal in specific energy. Do not confuse them with the conjugate (sequent) depths of a hydraulic jump, which are equal in specific force (momentum) and dissipate energy between them.
Source: Chow, V.T. (1959), Open-Channel Hydraulics, Ch. 3. Henderson, F.M., Open Channel Flow.
Worked example: critical depth and alternate depth
Rectangular channel, b = 8 ft, Q = 120 cfs, flowing at y = 3.0 ft
Both depths carry 120 cfs with 3.388 ft of specific energy. The channel slope and the controls determine which one occurs. The alternate-depth idea explains what happens at a sluice gate or a step: the depth jumps from one branch of the specific-energy curve to the other with almost no energy loss. A hydraulic jump goes the other way, from supercritical to subcritical, and dissipates energy. The two jump depths are therefore not alternate depths.
Common mistakes
- Using yc = (q²/g)1/3 in a trapezoidal or circular channel. That form is rectangular only. Other shapes need Q²T/(gA³) = 1 solved for y.
- Computing the Froude number with the hydraulic radius. Fr uses hydraulic depth D = A/T. The two coincide only in a wide rectangular channel.
- Assuming the flow passes through critical depth at a drop. At a free overfall, the brink depth is about 0.715 yc. Critical depth occurs a short distance upstream.
- Running a water-surface profile from the wrong end. Subcritical profiles start at a downstream control. Supercritical profiles start upstream. Starting from the wrong boundary gives a profile that looks plausible but is wrong.
When this card doesn’t apply
Specific energy assumes a hydrostatic pressure distribution and a uniform velocity, with α = 1. Steep channels need a cosθ correction to the depth. Strongly curved flow over a spillway crest or through a bend is not hydrostatic. Compound channels with floodplains can have more than one critical depth, which a single-section calculation will miss. For natural streams, use a step-backwater model such as HEC-RAS.
Related cheat sheets and tools
Compare critical depth against the normal depth from Manning's using the channel geometry card; if normal < critical the channel is steep (supercritical) and a hydraulic jump may form on a slope break. The gradually varied flow calculator steps the profile through that break, and the stilling basin calculator sizes a USBR basin where the jump is forced at a structure. For full water-surface-profile and routing analysis, see HydroComplete.